Mayor Thomas M. Menino remains overwhelmingly popular in Boston After nearly two decades in office, with almost three out of four residents saying they believe the city remains on the right track, according to findings from a Boston Globe poll. But residents appear concerned about Menino’s health and seem divided about whether he should run for a sixth term this fall.

Menino would seem to be in a commanding position to win reelection, with a 74 percent approval rating according to the Globe poll. Construction cranes dot Boston’s skyline, from Dudley Square to the Seaport, and roughly 65 percent of those polled said development has made Boston a better place to live. And almost half said they had met Menino personally, an extraordinary feat for a big city mayor.

Menino would trounce City Councilor John R. Connolly, who is seeking to unseat Menino, in a head-to-head match, 50 percent to 21 percent, with more than a quarter of voters undecided.

But within that deep vein of good will among voters, a more nuanced political view emerges. Fewer than half the respondents, 43 percent, said they wanted Menino to run for reelection versus 36 percent who did not. That sentiment — a fondness for Menino and his stewardship of the city, but a belief that a sixth term might be too much — could create an opening for a challenger.

The poll, conducted for the Globe by the Survey Center at the University of New Hampshire, comes at a critical time for Menino, 70, who has said he will decide soon whether to embark on another campaign.

“If Menino ran again, he would probably win handily, but I think there’s a desire among a lot of people to have a new mayor,” said Andrew E. Smith, director of the UNH Survey Center. “It’s not that they are unhappy with Menino. voters just think it’s time for new blood at the top.”

Menino is not in as strong of a position as he was in 2009, when a poll found him leading challenger Michael F. Flaherty Jr. 61 percent to 23 percent.

Menino has endured a series of medical problems in his current term, most significantly since October. A string of aliments left Menino hospitalized for eight weeks with blood clots and an acute respiratory infection. Doctors also diagnosed the mayor with type 2 diabetes. Menino emerged from the hospital with limited mobility, which could complicate another campaign.

The poll found that 60 percent said Menino’s poor health was a concern that could influence whether they voted for him.

Peter Scholnick, 63, participated in the poll and said in a follow-up interview that he has voted for Menino several times and generally holds the mayor in high regard. But the South End resident said he had not decided whether he would vote for Menino if he were to run.

“He spent two months in the hospital, and he can’t get around very well,” Scholnick said. “He’s old, and I’d rather see someone younger and more vibrant.”

But Jillian Greenwood, 29, of West Roxbury, dismissed such concerns.

“His experience is worth more than any health issues that he might have,” Greenwood said in a follow-up interview. “I think he’s really brought the city together.”

The poll also found that Menino’s experience can be a negative. Roughly 45 percent said the fact that Menino had already served five terms made it less likely they would vote for him if he ran for a sixth term.

The poll of 440 adults was conducted with live telephone interviews from March 20 to Tuesday. It included calls to both land lines and cellphones with 337 likely voters. The results were weighted by gender, race, neighborhood, and other factor

s. the poll has a margin of error of plus or minus 4.7 percentage points.

Menino must decide by 5 p.m. May 13 whether to apply for nomination papers, the first step in getting his name on the ballot for the preliminary election, scheduled for Sept. 24. The top two vote-getters will compete Nov. 5.

the poll found Connolly had room to grow with voters. Almost 36 percent said they had a favorable impression of Connolly, which Smith said was a good number. (Menino was viewed favorably by 73 percent.)

But many others indicated they knew little about the councilor from West Roxbury.

“The big thing for him,” Smith said, “is more than 50 percent of people don’t know him at all or don’t know enough to say whether they like him or dislike him.”

The poll found that 73 percent of adults believe Boston is on the right track, a jump of nearly 11 percentage points from May 2009, when Menino was preparing for his most recent reelection bid. At the same time, Menino’s standing dropped in another category: The percentage of people who said they had met the mayor fell to 49 percent, down eight points from four years ago.

Smith said that may reflect Menino’s having spent less time in the public eye and not having announced whether he will run. That may explain why there are more undecideds than this time four years ago.

But undecided doesn’t mean voters are anti-Menino.

“It’s not like that vote is going to Connolly,” Smith said. “It’s just parked on the sidelines, waiting to see what’s going to happen.”